The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.     

Visual spatial memory is enhanced in female rats (but inhibited in males) by dietary soy phytoestrogens

Background  

In learning and memory tasks, requiring visual spatial memory (VSM), males exhibit superior performance to females (a difference attributed to the hormonal influence of estrogen). This study examined the influence of phytoestrogens (estrogen-like plant compounds) on VSM, utilizing radial arm-maze methods to examine varying aspects of memory. Additionally, brain phytoestrogen, calbindin (CALB), and cyclooxygenase-2 (COX-2) levels were determined.     

The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds

Let \(G{/}H\) be a compact homogeneous space, and let \(\hat{g}_0\) and \(\hat{g}_1\) be G-invariant Riemannian metrics on \(G/H\). We consider the problem of finding a G-invariant Einstein metric g on the manifold \(G/H\times [0,1]\) subject to the constraint that g restricted to \(G{/}H\times \{0\}\) and \(G/H\times \{1\}\) coincides with \(\hat{g}_0\) and \(\hat{g}_1\), respectively. By assuming that the isotropy representation of \(G/H\) consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.     

Evaluation of glued-diaphragm fibre optic pressure sensors in a shock tube

Glued-diaphragm fibre optic pressure sensors that utilize standard telecommunications components which are based on Fabry–Perot interferometry are appealing in a number of respects. Principally, they have high spatial and temporal resolution and are low in cost. These features potentially make them well suited to operation in extreme environments produced in short-duration high-enthalpy wind tunnel facilities where spatial and temporal resolution are essential, but attrition rates for sensors are typically very high. The sensors we consider utilize a zirconia ferrule substrate and a thin copper foil which are bonded together using an adhesive. The sensors show a fast response and can measure fluctuations with a frequency up to 250 kHz. The sensors also have a high spatial resolution on the order of 0.1 mm. However, with the interrogation and calibration processes adopted in this work, apparent errors of up to 30% of the maximum pressure have been observed. Such errors are primarily caused by mechanical hysteresis and adhesive viscoelasticity. If a dynamic calibration is adopted, the maximum measurement error can be limited to about 10% of the maximum pressure. However, a better approach is to eliminate the adhesive from the construction process or design the diaphragm and substrate in a way that does not require the adhesive to carry a significant fraction of the mechanical loading.      

The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery–Hamel flow

In this paper Jeffery–Hamel flow has been studied and its nonlinear ordinary differential equation has been solved through homotopy analysis method (HAM). The obtained solution in comparison with the numerical ones represents a remarkable accuracy. The results also indicate that HAM can provide us with a convenient way to control and adjust the convergence region.     

Neuromodulation by soy diets or equol: Anti-depressive &; anti-obesity-like influences,age- &; hormone-dependent effects

Background  

Soy-derived isoflavones potentially protect against obesity and depression. In five different studies we examined the influence of soy-containing diets or equol injections on depression, serotonin levels, body weight gain (BW) and white adipose tissue (WAT) deposition in female Long-Evans rats at various stages of life [rats were intact, ovariectomized or experienced natural ovarian failure (NOF)].     

On the probability that given polynomials have a specified highest common factor

The polynomial functions f₁, f₂,…, f_m are found to have highest common factor h for a set of values of the variables x₁, x₂,…,x_m whose asymptotic density is

$\text{1}\text{h}{}^{\mathrm{n}}\text{∑}\text{d∣h}\text{μ(d)Π}{}^{\mathrm{m}}{}_{\mathrm{l\; =\; 1}}\text{?(f}{}_1\text{, dh)}\text{d}{}^{\mathrm{m}}\text{Π}\text{p∣h}\text{1?}\text{Π}{}^{\mathrm{m}}{}_{\mathrm{l\; =\; 1}}\text{?(f}{}_1\text{, p)}\text{p}{}^{\mathrm{m}}$

For the special case f₁(x) = f₂(x) = … = f_m(x) = x and h = 1 the above formula reduces to $\text{Π}{}_{\mathrm{?}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{m}}\text{) =}\text{1}\text{ζ}\text{(m)}$, the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f₁, f₂,…, f_m for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density.